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Eigenvalue distribution analysis of multidimensional prolate matrices

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Pith's one-line read This paper proves that the spectrum of a d-dimensional prolate matrix concentrates: up to a controlled error, (2MW)^d eigenvalues are ε-close to 1, at most C_d log(MW) log(1/ε) max{(log(MW) log(1/ε))^{d−1}, (2MW)^{d−1}} eigenvalues lie in…

desk verdict A legitimate d-dimensional extension of the 1D prolate eigenvalue bounds, but Theorem 1.1 as stated is false on its parameter range; the fix is straightforward. read the letter →

arxiv 2507.10412 v1 pith:YQBNVANT submitted 2025-07-14 math.CA

Luis Gomez , Jonathan Jaimangal , Azita Mayeli , Tasfia Proma This is my paper · ORCID
classification math.CA MSC 15A1842A3847B35
keywords prolatematriceseigenvalueconcentrationtime-frequencylocalizationtransitionbandtensorproductmultidimensionalDFTdiscretespheroidalsequencesnon-asymptoticbounds
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends the classical one-dimensional time–frequency concentration analysis of prolate matrices to signals defined on d-dimensional Cartesian grids. It proves a non-asymptotic eigenvalue-distribution theorem: for the multidimensional prolate matrix A = T_M B_K T_M, the number of eigenvalues above ε is pinned to (2MW)^d within an explicit error, and the width of the transition band between eigenvalues near 1 and near 0 is bounded by a quantified quantity involving log(MW), log(1/ε), and dimension d. If correct, this gives a rigorous quantitative version of the 'effective dimension' intuition: the number of jointly time-and-frequency-concentrated degrees of freedom in a d-dimensional discrete signal is approximately the time–bandwidth volume (2MW)^d, independent of the ambient grid size.

What carries the argument

The argument rests on three pieces: the multidimensional prolate matrix factors as a d-fold tensor product of one-dimensional prolate matrices, A = A_1 ⊗ ⋯ ⊗ A_d (Lemma 3.2), so its eigenvalues are products of 1D eigenvalues; a sandwiching lemma (Lemma 4.4) that relates the d-dimensional count of eigenvalues above ε to the 1D counts above ε and $ε^{{1/d}}$; and two cited one-dimensional results—the transition-region bound R_ε(MW) (Theorem 4.1, from reference [6]) and the 1/2-threshold ordering (Theorem 4.2, from reference [23])—which together yield Proposition 4.3, a sharp 1D estimate |#{λ > γ} − 2MW| ≲ R_γ(MW). Raising the 1D estimate to the d-th power and controlling the binomial error gives Lemma 4.6, which is the core of Theorem 1.1.

What would settle it

Compute the eigenvalue counts of the 2D prolate matrix numerically for, say, N = 100, M = 50, K = 20 (so W = 41/200 ≈ 0.205), and count eigenvalues in (0.1, 0.9). If n_{0.1} exceeds C_2 B_2(10.25, 0.1) for any apparent constant, or if m_ε deviates from (2MW)^2 = 420.25 more than the bound allows, the theorem fails. Equivalently, verify the 1D ordering λ_{⌊2MW⌋−1} ≥ 1/2 ≥ λ_{⌊2MW⌋+1} for small M and W near 1/2; any counterexample there would collapse Proposition 4.3.

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Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for every d ≥ 1, M < N, K ≤ ⌊(N−1)/2⌋, W = (2K+1)/(2N) ∈ (0,1/2), and ε > 0, the eigenvalue counts of the multidimensional prolate matrix satisfy |m_ε(M,K) − (2MW)^d| ≤ C_d B_d(MW,ε) and n_ε(M,K) ≤ C_d B_d(MW,ε), where B_d(MW,ε) = log(MW) log(1/ε) max{(log(MW) log(1/ε))^{d−1}, (2MW)^{d−1}} and C_d depends only on d. In words: the bulk of the eigenvalues cluster near 1 or 0, the number of eigenvalues that are ε-close to 1 is approximately (2MW)^d, and the transition band contains at most O((log(MW) log(1/ε))^d) eigenvalues.

Load-bearing premise

The proof assumes that the two cited one-dimensional estimates—the transition-region bound and the 1/2-threshold ordering—hold for every M ≤ N and W ∈ (0,1/2) with the stated error R_ε(MW); if either bound has hidden regime restrictions, the multidimensional result inherits them.

Editorial extensions

If this is right

  • For images and volumetric data, the number of significant degrees of freedom in a time–frequency window is quantitatively (2MW)^d, giving a principled dimension reduction for multidimensional signals.
  • The non-asymptotic error bound means the concentration holds for finite grids, not just in a limiting regime, so algorithms that threshold eigenvalues of prolate matrices in d dimensions have a proven accuracy guarantee.
  • The transition-band bound scales as O((log(MW) log(1/ε))^d), so the 'plunge region' stays narrow relative to the number of significant eigenvalues even as the time–bandwidth product grows.
  • The result extends the known 1D estimates of [6] to d dimensions, closing the gap identified in the literature for non-asymptotic higher-dimensional bounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same tensor-product sandwiching method likely extends to rectangular grids with different M_i and W_i per dimension, replacing (2MW)^d by ∏(2M_i W_i) with a sum of per-dimension error terms; the paper does not state this, but it follows from Lemma 3.2.
  • Tracking the dimension-dependent constant C_d carefully could turn Theorem 1.1 into a practical truncation rule for multidimensional DPSS-based compression, since the bulk eigenvalue count and the transition width are both explicit.
  • Proposition 3.4, which counts zeros of the multidimensional Dirichlet kernel, hints at a combinatorial counterpart to eigenvalue counting in higher dimensions; a testable extension would compare the transition-band width to zero-crossing counts of the product kernel.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper defines d-dimensional discrete prolate matrices as tensor products of one-dimensional time- and band-limiting projections, proves that their eigenvalues are products of one-dimensional prolate eigenvalues, and derives non-asymptotic eigenvalue-distribution bounds. The main claim, Theorem 1.1, states that for every epsilon>0 the number m_epsilon of eigenvalues above epsilon is within C_d B_d(MW,epsilon) of (2MW)^d, and that the number n_epsilon of eigenvalues in the transition band (epsilon,1-epsilon) is at most C_d B_d(MW,epsilon), where B_d involves log(MW)log(1/epsilon). The proof reduces the d-dimensional counting to known one-dimensional bounds of Karnik et al. and Zhu-Wakin via product-counting lemmas.

Significance. If the theorem is corrected, the paper would provide a useful multidimensional discrete analogue of the classical Slepian eigenvalue-concentration phenomenon, with explicit dimension-dependent transition bounds. The tensor-product reduction in Lemma 3.2 and the product-counting argument in Lemmas 4.4-4.6 are clean and give a credible strategy. The paper also makes a welcome effort to track constants and non-asymptotic dependence on the time-bandwidth product and on epsilon. However, the main theorem as stated is false on part of its declared parameter range, and the proof contains invalid estimates for small MW and for epsilon>1/2. These issues are localized and appear fixable, but they currently block acceptance.

major comments (3)
  1. [Theorem 1.1, Eq. (1.3); Lemma 4.6, Eq. (4.2)] The bound B_d(MW,epsilon) is not a valid upper bound when MW<1, because log(MW) is negative while the left-hand side is nonnegative. This is not a vacuous restriction: for d=1, N=100, M=5, K=0, we have W=1/200 and MW=0.025, and A=T_M B_K T_M is the 5x5 matrix with all entries 1/100. Its eigenvalues are 0.05,0,0,0,0, so for epsilon=0.01 we have m_epsilon=1 and |m_epsilon-(2MW)^d|=0.95, whereas B_1(0.025,0.01)=log(0.025)log(100)<0. Thus (1.3) cannot hold for any C_1. Since MW=M(2K+1)/(2N) can be arbitrarily small, the theorem must either assume MW>=1 or replace log(MW) by a positive majorant such as log(100MW+25).
  2. [Theorem 1.1, Eq. (1.3); Lemma 4.6, final paragraph] The first inequality of Theorem 1.1 is stated for epsilon in (0,1), but the proof only establishes the estimate for epsilon in (0,1/2). Lemma 4.5(i), which gives chi(epsilon)<=2log(1/epsilon), is valid only for epsilon<=1/2, and the final paragraph of Lemma 4.6 attempts to extend the result to epsilon>1/2 using m_epsilon <= m_{1-epsilon}. That argument is invalid because it imports the nonsymmetric factor log(1/epsilon) into the right-hand side: as epsilon approaches 1, the right side of (1.3) tends to 0 while the left side equals (2MW)^d once epsilon exceeds the largest eigenvalue. The theorem should either restrict the first estimate to epsilon in (0,1/2) or use the symmetric quantity chi(epsilon)=log(1/(epsilon(1-epsilon))) in B_d.
  3. [Lemma 4.6, proof around Eq. (4.4)] The proof replaces the one-dimensional bound R_epsilon(MW) from Theorem 4.1 by the assertion R_epsilon(MW) \lesssim log(MW)chi(epsilon). This is false for MW<1 because Theorem 4.1 gives R_epsilon(MW)=(2/pi^2)log(100MW+25)log(5/(epsilon(1-epsilon)))+7, which is positive and at least 7, while log(MW) tends to -infinity as MW tends to 0. This is precisely the step that produces the negative or zero right-hand side in the counterexample above. A correct proof needs a positive majorant such as log(100MW+25), with an additive constant handled explicitly.
minor comments (4)
  1. [Section 3.0.1, Proposition 3.4] Proposition 3.4 appears to be incorrect as stated: the estimate #Z_W=2(M-r) with r approximately MW is inconsistent with the concluding bound |#Z_W-2floor(MW)|<=2, and the proof counts zeros by treating every j with j/(2W)<=M-1 as a zero, although one must additionally require 2WDelta to be an integer. For example, with W=1/4 and M=100, the proof would give #Z_W=98 while 2floor(MW)=50. Since this proposition is not used in the proof of Theorem 1.1, it does not affect the main result, but it should be corrected or removed.
  2. [Proposition 4.3, last line] The last line of the proof of Proposition 4.3 contains the typo '2NW' where '2MW' is intended; the printed statement otherwise uses M and W consistently.
  3. [References [22] and [23]] References [22] and [23] list the same paper by Zhu and Wakin with different page ranges; Theorem 4.2 cites [23] while the related-work discussion cites [22], which will confuse readers tracking the one-dimensional threshold result.
  4. [Lemma 2.7] The displayed formula in Lemma 2.7, '(B_K x)[n] = prod_i (B_{K_i})(n_i)', is not a meaningful operator identity as written; it should either be stated for separable inputs x=x_1 tensor ... tensor x_d or be replaced by the corresponding entrywise product formula for the kernel.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the d-dimensional eigenvalue-counting argument rests on externally cited 1D bounds and a self-contained tensor-product proof.

full rationale

The paper's central estimate in Theorem 1.1 is derived from two one-dimensional results that are cited from external authors: the transition-region bound of Karnik et al. (Theorem 4.1, reference [6]) and the 1/2-threshold ordering of Zhu and Wakin (Theorem 4.2, reference [23]). Neither result is produced by the present authors, and they are not derived from the multidimensional claim being proved. The multidimensional extension is built on a tensor-product spectral decomposition (Lemma 3.2), which is proven directly in the paper, and on counting arguments (Lemma 4.4, Proposition 4.3) that combine the 1D inputs with elementary inequalities. The only self-citation is to the third author's earlier work [5] (Israel–Mayeli), and the paper explicitly says it borrows techniques from that paper, but the main theorem does not rely on [5] as a source of the eigenvalue-counting estimates. The cited 1D bounds are independent, published results with stated assumptions that do not include the target d-dimensional result, and the new counting step is a separate argument. No parameter is fitted and no known result is merely renamed. Consequently, there is no significant circularity; the derivation chain is externally supported and self-contained apart from standard citations.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper contributes no fitted parameters and introduces no new entities. It relies on three cited 1D results: the Karnik et al. transition bound, the Zhu-Wakin eigenvalue ordering, and Slepian's eigenvalue bounds. The d-dimensional product-counting theorem is then derived from these.

assumptions (3)
  • domain assumption The 1D transition-region bound of Karnik, Romberg, and Davenport (Theorem 4.1, reference [6]) states that for all M ≤ N and W ∈ (0,1/2), the number of eigenvalues of the 1D prolate matrix in (ε, 1-ε) is at most (2/π^2) log(100MW+25) log(5/(ε(1-ε))) + 7.
    Invoked in Section 4.1 as Theorem 4.1; the d-dimensional transition-band bound is obtained by combining it with product counting.
  • domain assumption The 1D eigenvalue ordering of Zhu and Wakin (Theorem 4.2, reference [23]) states that λ_{⌊2MW⌋-1} ≥ 1/2 ≥ λ_{⌊2MW⌋+1}, meaning about 2MW eigenvalues lie above the 1/2 threshold.
    Invoked in Section 4.1 as Theorem 4.2; Proposition 4.3 uses it to identify the number of eigenvalues near 1 before applying the transition bound.
  • domain assumption Slepian's 1D result (reference [18]) that the eigenvalues of the 1D prolate matrix are strictly between 0 and 1 and non-degenerate.
    Used in Theorem 3.3 to conclude the d-dimensional eigenvalues lie in (0,1) and in setting up Lemma 3.2's product spectrum.

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Pith. "Pith review of Eigenvalue distribution analysis of multidimensional prolate matrices." pith.science (2026). https://pith.science/paper/YQBNVANT

@misc{pith2026250710412,
  author       = {Pith},
  title        = {Pith review of: Eigenvalue distribution analysis of multidimensional prolate matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQBNVANT}},
  note         = {Machine review of arXiv:2507.10412}
}
read the original abstract

We extend classical time-frequency limiting analysis, historically applied to one-dimensional finite signals, to the multidimensional discrete setting. This extension is relevant for images, videos, and other multidimensional signals, as it enables a rigorous study of joint time-frequency localization in higher dimensions. To achieve this, we define multidimensional time-limiting and frequency-limiting matrices tailored to signals on a Cartesian grid and construct a multi-indexed prolate matrix. We prove that the spectrum of this matrix exhibits an eigenvalue concentration phenomenon: the bulk of eigenvalues cluster near 1 or 0 with a narrow transition band separating these regions. Moreover, we derive quantitative bounds on the width of the transition band in terms of the time-bandwidth product and prescribed accuracy. Concretely, our contributions are twofold: (i) we extend existing one-dimensional results to higher-dimensional Cartesian discrete signals; and (ii) we develop a multidimensional non-asymptotic eigenvalue-distribution analysis for prolate matrices. The advances are summarized in Theorem 1.1. Numerical experiments in one- and two-dimensional settings confirm the predicted eigenvalue concentration and illustrate potential applications in fast computation for image analysis, multidimensional spectral estimation, and related signal-processing tasks.

Figures

Figures reproduced from arXiv: 2507.10412 by the authors.

Figure 3
Figure 3. In [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 1
Figure 1. Influence of time-bandwidth bound- Eigenvalue spectra of time– frequency limiting (prolate) matrices for increasing ambient dimension N, while keeping the time-bandwidth product 2MW fixed. Each subplot shows the first 100 eigenvalues of the prolate matrix A = TMBKTM, where BK is the frequency￾limiting operator to bandwidth K, and TM is the time-limiting operator to M samples. Despite increasing N, the number of sign… view at source ↗
Figure 2
Figure 2. In all these graphs, the time-bandwidth is 2MW ≈ 270. From left to right, the plots correspond to W ≈ 0.45, W ≈ 0.22, and W ≈ 0.15. This shows that the eigenvalue distribution depends primarily on the product MW, and remains consistent across different N, M and W configurations as long as MW held constant [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figures from the paper (3 more)
Figure 3
Figure 3. Figure 3: Eigenvalue distributions of prolate matrices A = TMBKTM for in￾creasing ambient dimension N, and parameters K and M, while keeping the time￾bandwidth product 2MW fixed. Each subplot shows the first 100 eigenvalues. As N increases (from left to right), the shape and dec…
Figure 4
Figure 4. Figure 4: This graph shows the plot of eignvalues λ (ℓ) N as a function of MW or W for a fixed M. Here, we choose N = 1000, M = 800, and allow W and 2MW to vary over the intervals (0, 0.2) and (0, 400), respectively. The graphs from left to write show λ (0) 1000, λ (1) 1000, λ(2…
Figure 5
Figure 5. Figure 5: Eigenvalue distribution of a one-dimensional prolate matrix and its tensor product. The left plot shows the eigenvalues of a 1D prolate matrix A = TMBKTM, with time-bandwidth product of 2WM = 5. As shown, the leading eigenvalues are close to 1, while the remaining ones…

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Works this paper leans on

23 extracted references · 22 canonical work pages

  1. [1]

    Discrete prolate spheroidal wave func- tions: Further spectral analysis and some related applications

    M. Boulsane, N. Bourguiba, and A. Karoui. “Discrete prolate spheroidal wave func- tions: Further spectral analysis and some related applications”. In:Journal of Scientific Computing 82.3 (2020), p. 54

    work page 2020

  2. [2]

    Time-Frequency Localization Operators: A Geometric Phase Space Approach

    I. Daubechies. “Time-Frequency Localization Operators: A Geometric Phase Space Approach”. In: IEEE Transactions on Information Theory 34.4 (1988), pp. 605–612

    work page 1988

  3. [3]

    On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II

    K. Hughes, A. Israel, and A. Mayeli. “On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II”. In: to appear in JF AA, arXiv:2403.13092 (2024)

    work page Pith review arXiv 2024

  4. [4]

    The eigenvalue distribution of time-frequency localization operators

    A. Israel. “The eigenvalue distribution of time-frequency localization operators”. In: arXiv preprint arXiv:1502.04404 (2015)

    arXiv 2015

  5. [5]

    On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions

    A. Israel and A. Mayeli. “On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions”. In: Applied and Computational Harmonic Analysis 70 (2024), p. 101620

    work page 2024

  6. [6]

    Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences

    S. Karnik, J. Romberg, and M. Davenport. “Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences”. In: Ap- plied and Computational Harmonic Analysis 55.1 (2021), pp. 97–128

    work page 2021

  7. [7]

    The fast Slepian transform

    S. Karnik et al. “The fast Slepian transform”. In: Applied and Computational Harmonic Analysis 46.3 (2019), pp. 624–652

    work page 2019

  8. [8]

    Approximation scheme for essentially bandlimited and space-concentrated functions on a disk

    B. Landa and Y. Shkolnisky. “Approximation scheme for essentially bandlimited and space-concentrated functions on a disk”. In: Applied and Computational Harmonic Analysis 43.3 (2017), pp. 381–403

    work page 2017

Show all 23 references
  1. [9]

    Steerable principal components for space-frequency lo- calized images

    B. Landa and Y. Shkolnisky. “Steerable principal components for space-frequency lo- calized images”. In: SIAM J. Imaging Sci. 10.2 (2017), pp. 508–534

    2017

  2. [10]

    Prolate spheroidal wave functions, Fourier analysis, and uncertainty – II

    H.J. Landau and H.O. Pollak. “Prolate spheroidal wave functions, Fourier analysis, and uncertainty – II”. In: Bell Systems Tech. J. 40.1 (1961), pp. 65–84

    1961

  3. [11]

    Prolate spheroidal wave functions, Fourier analysis, and uncertainty – III : The dimension of the space of essentially time- and band-limited signals

    H.J. Landau and H.O. Pollak. “Prolate spheroidal wave functions, Fourier analysis, and uncertainty – III : The dimension of the space of essentially time- and band-limited signals”. In: Bell Systems Tech. J. 41.4 (1962), pp. 1295–1336

    1962

  4. [12]

    Eigenvalue distribution of time and frequency limiting

    H.J. Landau and H. Widom. “Eigenvalue distribution of time and frequency limiting”. In: Journal of Mathematical Analysis and Applications 77.2 (1980), pp. 469–481

    1980

  5. [13]

    A representation theory perspective on simultaneous alignment and classification

    R. Lederman and A. Singer. “A representation theory perspective on simultaneous alignment and classification”. In: Applied and Computational Harmonic Analysis 49.3 (2020), pp. 1001–1024

    2020

  6. [14]

    Continuously heterogeneous hyper-objects in cryo-EM and 3-D movies of many temporal dimensions

    R.R. Lederman and A. Singer. “Continuously heterogeneous hyper-objects in cryo-EM and 3-D movies of many temporal dimensions”. In: arXiv preprint arXiv:1704.02899 (2017)

    2017 arXiv

  7. [15]

    Numerical algorithms for the computation of generalized prolate spheroidal functions

    Roy R Lederman. “Numerical algorithms for the computation of generalized prolate spheroidal functions”. In: arXiv preprint arXiv:1710.02874 (2017). REFERENCES 24

    2017 arXiv

  8. [16]

    Eigenvalue estimates for Fourier concentration operators on two domains

    F. Marceca, Jos´ e Luis Romero, and M. Speckbacher. “Eigenvalue estimates for Fourier concentration operators on two domains”. In:Archive for Rational Mechanics and Anal- ysis 248.3 (2024), p. 35

    2024

  9. [17]

    Prolate spheroidal wave functions, Fourier analysis, and uncertainty – V: The discrete case

    D. Slepian. “Prolate spheroidal wave functions, Fourier analysis, and uncertainty – V: The discrete case”. In: Bell Systems Tech. J. 57.5 (1978), pp. 1371–1430

    1978

  10. [18]

    Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V- The discrete case

    D. Slepian. “Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V- The discrete case”. In: Bell Systems Tech. J. 57.5 (1978), pp. 1371–1430

    1978

  11. [19]

    Prolate spheroidal wave functions, Fourier analysis, and uncertainty – I

    D. Slepian and H.O. Pollak. “Prolate spheroidal wave functions, Fourier analysis, and uncertainty – I”. In: Bell Systems Tech. J. 40.1 (1961), pp. 43–64

    1961

  12. [20]

    H. Widom. Asymptotic expansions for pseudodifferential operators on bounded domains. Vol. 1152. Springer, 2006

    2006

  13. [21]

    Two dimensional prolate spheroidal wave functions for MRI

    Q.X. Yang et al. “Two dimensional prolate spheroidal wave functions for MRI”. In: Journal of Magnetic Resonance 158 (2002), pp. 43–51

    2002

  14. [22]

    Approximating sampled sinusoids and multiband signals using multiband modulated DPSS dictionaries

    Z. Zhu and M. B. Wakin. “Approximating sampled sinusoids and multiband signals using multiband modulated DPSS dictionaries”. In: J. Fourier Anal. Appl. 23.6 (2017), pp. 1263–1310

    2017

  15. [23]

    Approximating sampled sinusoids and multiband signals using multiband modulated DPSS dictionaries

    Z. Zhu and M. B. Wakin. “Approximating sampled sinusoids and multiband signals using multiband modulated DPSS dictionaries”. In: Journal of Fourier Analysis and Applications 23 (2017), pp. 1263–1310. REFERENCES 25 Appendix A. Figures Figure 1: Influence of time-bandwidth bound...

    2017

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